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Finite Element Modeling of Microstructures in Composite Materials

A special issue of Materials (ISSN 1996-1944). This special issue belongs to the section "Advanced Composites".

Deadline for manuscript submissions: closed (10 March 2024) | Viewed by 8002

Special Issue Editor


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Guest Editor
Department of Mechanical Engineering, University of Manitoba, Winnipeg, MB, Canada
Interests: finite element modeling; micromechanics of composite materials; design and analysis of functionally graded materials; multiscale modeling; composite strength and fracture

Special Issue Information

Dear Colleagues,

Composite microstructures, including both composition and geometry, play a crucial role in the regulation of composite macroscopic properties. Fully understanding the relationship between composite microstructure and macroscopic properties is the fundamental base for the effective design of novel composites. However, this relationship is so complex that, although it has been extensively explored, there are still many aspects that are either unclear or completely unknown. Although finite element modeling is believed to be a more efficient approach than analytical and experimental methods for further the understanding of this relationship, it also faces a number of challenges. For this Special Issue, we invite high-quality papers showing recent progress in addressing these challenges. Topics of interest include, but are not limited to:

  • Finite element models of composite microstructure validated by experiments;
  • Analytical formulas established from finite element modeling;
  • Design and analysis of functionally graded materials;
  • Multiscale modeling of composite microstructure;
  • Relation between composite nonlinear behavior and microstructural local damage;
  • Finite element modeling of 3D-printed composites.

All submissions will undergo a rigorous peer-reviewing process.

Prof. Dr. Yunhua Luo
Guest Editor

Manuscript Submission Information

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Keywords

  • composite material
  • microstructure
  • finite element modeling
  • macroscopic property
  • local damage
  • functionally graded material
  • 3D-printed composite
  • multiscale modeling

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Published Papers (5 papers)

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Editorial

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2 pages, 157 KiB  
Editorial
Finite Element Modeling of Microstructures in Composite Materials: A Special Issue in Materials
by Yunhua Luo
Materials 2023, 16(15), 5332; https://doi.org/10.3390/ma16155332 - 29 Jul 2023
Viewed by 1273
Abstract
This Special Issue of the journal Materials aims to gather recent advancements and novel developments in the field of finite element modeling of microstructures in composite materials [...] Full article
(This article belongs to the Special Issue Finite Element Modeling of Microstructures in Composite Materials)

Research

Jump to: Editorial

19 pages, 3945 KiB  
Article
A Novel Finite Element-Based Method for Predicting the Permeability of Heterogeneous and Anisotropic Porous Microstructures
by Paris Mulye, Elena Syerko, Christophe Binetruy and Adrien Leygue
Materials 2024, 17(12), 2873; https://doi.org/10.3390/ma17122873 - 12 Jun 2024
Cited by 2 | Viewed by 1270
Abstract
Permeability is a fundamental property of porous media. It quantifies the ease with which a fluid can flow under the effect of a pressure gradient in a network of connected pores. Porous materials can be natural, such as soil and rocks, or synthetic, [...] Read more.
Permeability is a fundamental property of porous media. It quantifies the ease with which a fluid can flow under the effect of a pressure gradient in a network of connected pores. Porous materials can be natural, such as soil and rocks, or synthetic, such as a densified network of fibres or open-cell foams. The measurement of permeability is difficult and time-consuming in heterogeneous and anisotropic porous media; thus, a numerical approach based on the calculation of the tensor components on a 3D image of the material can be very advantageous. For this type of microstructure, it is important to perform calculations on large samples using boundary conditions that do not suppress the transverse flows that occur when flow is forced out of the principal directions. Since these are not necessarily known in complex media, the permeability determination method must not introduce bias by generating non-physical flows. A new finite element-based method proposed in this study allows us to solve very high-dimensional flow problems while limiting the biases associated with boundary conditions and the small size of the numerical samples addressed. This method includes a new boundary condition, full permeability tensor identification based on the multiscale homogenization approach, and an optimized solver to handle flow problems with a large number of degrees of freedom. The method is first validated against academic test cases and against the results of a recent permeability benchmark exercise. The results underline the suitability of the proposed approach for heterogeneous and anisotropic microstructures. Full article
(This article belongs to the Special Issue Finite Element Modeling of Microstructures in Composite Materials)
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Figure 1

Figure 1
<p>Overview and scope of PoroS: The input can be either a segmented 3D scan of the material or a digital twin of the material generated using TexGen<sup>®</sup>. The scope of PoroS consists of the Stokes flow problem solver and a full-field homogenization subroutine. Thus, the output of the solver consists of flow fields of the flow problems and a full 3D permeability tensor.</p>
Full article ">Figure 2
<p>(<b>a</b>) Poiseuille flow in a 3D pipe with a circular cross section. (<b>b</b>) Cross sectional view of the pipe. (<b>c</b>) Comparison of the <math display="inline"><semantics> <msub> <mi>V</mi> <mi>x</mi> </msub> </semantics></math> profile in the middle of the domain obtained using PoroS with the known analytical solution.</p>
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<p>(<b>a</b>) Magnitude of velocity field in [<math display="inline"><semantics> <mi mathvariant="sans-serif">μ</mi> </semantics></math>m/s] in case of a body force-driven flow (<math display="inline"><semantics> <mrow> <msub> <mi>V</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>0.493</mn> </mrow> </semantics></math>). (<b>b</b>) Comparison of the normalized transverse permeability (<math display="inline"><semantics> <mrow> <msub> <mi>K</mi> <mrow> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo>/</mo> <msup> <mi>R</mi> <mn>2</mn> </msup> </mrow> </semantics></math>) obtained using an analytical expression from [<a href="#B30-materials-17-02873" class="html-bibr">30</a>], numerical simulations performed using a body force-driven flow, and numerical simulations performed using a Dirichlet condition-driven flow.</p>
Full article ">Figure 4
<p>Geometry of the models for the calculation of the longitudinal permeability: a square channel of cross section: (<b>a</b>) <math display="inline"><semantics> <mrow> <mn>20</mn> <mo>×</mo> <mn>20</mn> </mrow> </semantics></math> [<math display="inline"><semantics> <mi mathvariant="sans-serif">μ</mi> </semantics></math>m]; (<b>b</b>) <math display="inline"><semantics> <mrow> <mn>40</mn> <mo>×</mo> <mn>40</mn> </mrow> </semantics></math> [<math display="inline"><semantics> <mi mathvariant="sans-serif">μ</mi> </semantics></math>m]; (<b>c</b>) <math display="inline"><semantics> <mrow> <mn>60</mn> <mo>×</mo> <mn>60</mn> </mrow> </semantics></math> [<math display="inline"><semantics> <mi mathvariant="sans-serif">μ</mi> </semantics></math>m]; (<b>d</b>) <math display="inline"><semantics> <mrow> <mn>80</mn> <mo>×</mo> <mn>80</mn> </mrow> </semantics></math> [<math display="inline"><semantics> <mi mathvariant="sans-serif">μ</mi> </semantics></math>m].</p>
Full article ">Figure 5
<p>(<b>a</b>) The 3D segmented image used as an input for the international virtual permeability benchmark. (<b>b</b>) Comparison of results obtained with PoroS solver with body forcing with respect to all the results of the benchmark participants.</p>
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<p>Comparison with the benchmark results: transverse sections (<b>a</b>); <math display="inline"><semantics> <msub> <mi>K</mi> <mrow> <mi>x</mi> <mi>x</mi> </mrow> </msub> </semantics></math> (<b>b</b>); <math display="inline"><semantics> <msub> <mi>K</mi> <mrow> <mi>y</mi> <mi>y</mi> </mrow> </msub> </semantics></math> (<b>c</b>); <math display="inline"><semantics> <msub> <mi>K</mi> <mrow> <mi>z</mi> <mi>z</mi> </mrow> </msub> </semantics></math> (<b>d</b>).</p>
Full article ">Figure 7
<p>Comparison with the benchmark results: (<b>a</b>) longitudinal sections; (<b>b</b>) <math display="inline"><semantics> <msub> <mi>K</mi> <mrow> <mi>y</mi> <mi>y</mi> </mrow> </msub> </semantics></math>.</p>
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<p>(<b>a</b>) Geometry of a 2D channel network inclined at <math display="inline"><semantics> <msup> <mn>60</mn> <mo>∘</mo> </msup> </semantics></math>. (<b>b</b>) Voxelized zoomed view of geometry.</p>
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<p>Field of <math display="inline"><semantics> <mrow> <mo>|</mo> <mi mathvariant="bold-italic">V</mi> <mo>|</mo> </mrow> </semantics></math> in [<math display="inline"><semantics> <mi mathvariant="sans-serif">μ</mi> </semantics></math>m/s] for (<b>a</b>) flow problem in X and (<b>b</b>) flow problem in Y.</p>
Full article ">Figure 10
<p>Local fibres’ misalignment changing the flow principal directions in the case of the longitudinally cut sub-volume 10 (refer to <a href="#materials-17-02873-f007" class="html-fig">Figure 7</a>a) from the international virtual permeability benchmark case.</p>
Full article ">
18 pages, 8768 KiB  
Article
The Tensile Behavior of Hybrid Bonded Bolted Composite Joints: 3D-Digital Image Correlation versus Finite Element Analysis
by Raphael Blier, Leila Monajati, Masoud Mehrabian and Rachid Boukhili
Materials 2024, 17(7), 1675; https://doi.org/10.3390/ma17071675 - 5 Apr 2024
Cited by 2 | Viewed by 1650
Abstract
This study examines the behavior of hybrid bolted/bonded (HBB) joints loaded in tensile shear comprising plain weave carbon/epoxy laminates in quasi-isotropic (QI) and cross-ply (CP) layups. It proposes a combined approach of 3D digital image correlation and finite element analysis (FEA) to assess [...] Read more.
This study examines the behavior of hybrid bolted/bonded (HBB) joints loaded in tensile shear comprising plain weave carbon/epoxy laminates in quasi-isotropic (QI) and cross-ply (CP) layups. It proposes a combined approach of 3D digital image correlation and finite element analysis (FEA) to assess their behavior. To apply the FEA simulation accurately, a single layer of plain fabric was replaced with [0/90]s lamination. Experimental standard open-hole tension test results, as well as only bolted (OB) and HBB, along with FEA predictions, confirmed the accuracy of the substitution method. The FEA, calibrated by experimental results, provides insight into the distinctive characteristics of HBB joints in comparison with bonded and bolted joints. Critical considerations include material properties, damage modeling, adhesive characteristics, and mass scaling. The FEA results underscored the pivotal role of adhesives in HBB joints, rendering them akin solely to bonded configurations. HBB joints retain their geometry better than OB joints with considerably less out-of-plane displacement, following a sinusoidal trend. Moreover, the overall behavior of the two layups demonstrates that CP benefits from having higher strength than QI, especially at the critical hole located closer to the grip side. Full article
(This article belongs to the Special Issue Finite Element Modeling of Microstructures in Composite Materials)
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Figure 1

Figure 1
<p>Specimen geometry and dimensions in mm for (<b>a</b>) OHT and (<b>b</b>) OB and HBB SL joints.</p>
Full article ">Figure 1 Cont.
<p>Specimen geometry and dimensions in mm for (<b>a</b>) OHT and (<b>b</b>) OB and HBB SL joints.</p>
Full article ">Figure 2
<p>Woven ply simplification.</p>
Full article ">Figure 3
<p>Damage initiation failure flow chart.</p>
Full article ">Figure 4
<p>Araldite<sup>®</sup> LY 8601/Aradur<sup>®</sup> 8602 epoxy system uniaxial tensile stress–strain curve.</p>
Full article ">Figure 5
<p>Linear traction separation cohesive zone model.</p>
Full article ">Figure 6
<p>Strain field in the loading direction ϵ<sub>xx</sub> at 25% of ultimate load for OHT CP12 and QI12 laminates comparing FEA with 3D-DIC results [<a href="#B14-materials-17-01675" class="html-bibr">14</a>].</p>
Full article ">Figure 7
<p>Nominal stress–displacement curves for OB and HBB Joints, case of CP layup.</p>
Full article ">Figure 8
<p>Nominal stress–displacement curves for OB and HBB Joints, case of QI layup.</p>
Full article ">Figure 9
<p>Longitudinal strain field for the CPlayup away from the washer; comparison between 3D-DIC results [<a href="#B1-materials-17-01675" class="html-bibr">1</a>] and simulation.</p>
Full article ">Figure 10
<p>Longitudinal strain field for the QI layup away from the washer; comparison between 3D-DIC results [<a href="#B1-materials-17-01675" class="html-bibr">1</a>] and simulation.</p>
Full article ">Figure 11
<p>Load–displacement curves for QI12 and CP12 layups.</p>
Full article ">Figure 12
<p>Comparison of stresses in the adhesive layer for QI12 and CP12 HBB joints.</p>
Full article ">Figure 13
<p>Comparison of the out-of-plane displacement for QI12 and CP12 HBB joints.</p>
Full article ">Figure 14
<p>OPD for different joining methods using CP12 at various load levels.</p>
Full article ">Figure 15
<p>Scaled OPD of OB and HBB joints at 8 KN, case of CP12 layup.</p>
Full article ">Figure 16
<p>Longitudinal strain field comparison using simulation depending on the joint configuration, case of CP layup.</p>
Full article ">Figure 17
<p>Longitudinal strain field comparison using simulation depending on the joint configuration, case of QI layup.</p>
Full article ">
20 pages, 7815 KiB  
Article
An Equivalent Structural Stress-Based Frequency-Domain Fatigue Assessment Approach for Welded Structures under Random Loading
by Uchenna Kalu and Xihui Liang
Materials 2023, 16(23), 7420; https://doi.org/10.3390/ma16237420 - 29 Nov 2023
Cited by 2 | Viewed by 1549
Abstract
Welded structures under random loadings are usually susceptible to fatigue-induced failures that lead to significant economic and safety effects. However, accurately predicting these structures’ fatigue damage and life in the frequency domain remains challenging due to the limitations associated with using traditional weld [...] Read more.
Welded structures under random loadings are usually susceptible to fatigue-induced failures that lead to significant economic and safety effects. However, accurately predicting these structures’ fatigue damage and life in the frequency domain remains challenging due to the limitations associated with using traditional weld stress extrapolation methods, such as nominal, hotspot, and notch stress methods. These methods struggle with precisely defining and characterizing the stresses at the weld toe and root as they vary depending on factors like weld stress concentration effects, joint geometry, and loading modes. This research introduces an Equilibrium Equivalent Structural Stress (EESS)-based frequency-domain fatigue analysis approach for welded structures subjected to random loading. The proposed method utilizes the EESS formulations, which are based on the decomposition and characterization of weld toe stresses with a single stress parameter, together with incorporating structural dynamic properties’ effects on the stresses acting on the weld joints and the corresponding accumulated fatigue damage of the structure. The numerical demonstration and validation of the proposed method have been performed using a welded Rectangular Hollow Section (RHS) T-joint structure subjected to stationary random fatigue loading. The proposed method’s fatigue damage and life results are compared with the fatigue test data and the equivalent hotspot stress extrapolation-based technique results. Full article
(This article belongs to the Special Issue Finite Element Modeling of Microstructures in Composite Materials)
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Figure 1

Figure 1
<p>Structural stress definition and calculation from finite element model mesh elements.</p>
Full article ">Figure 2
<p>Flowchart of the frequency-domain fatigue analysis method based on the EESS.</p>
Full article ">Figure 3
<p>Finite element model and boundary condition of the structure.</p>
Full article ">Figure 4
<p>Input base acceleration PSD load in the time-domain and frequency-domain form.</p>
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<p>Mode shapes contour and corresponding natural frequencies of the first ten modes of the structure.</p>
Full article ">Figure 6
<p>Evaluated modal EESS for the first ten modal responses of the structure along the weld path.</p>
Full article ">Figure 6 Cont.
<p>Evaluated modal EESS for the first ten modal responses of the structure along the weld path.</p>
Full article ">Figure 7
<p>Evaluated frequency response coordinates for the first ten modal responses of the structure.</p>
Full article ">Figure 8
<p>Maximum EESS FRF at node six on the weld toe paths.</p>
Full article ">Figure 9
<p>Maximum EESS response at node six: critical response at modes 2 and 10.</p>
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<p>PDF of EESS amplitudes for the maximum EESS PSD.</p>
Full article ">Figure 11
<p>PDF of fatigue damage intensity.</p>
Full article ">Figure 12
<p>Predicted fatigue damage intensity along the weld toe path.</p>
Full article ">
22 pages, 17392 KiB  
Article
Characterization and Design of Three-Phase Particulate Composites: Microstructure-Free Finite Element Modeling vs. Analytical Micromechanics Models
by Sebak Oli and Yunhua Luo
Materials 2023, 16(18), 6147; https://doi.org/10.3390/ma16186147 - 10 Sep 2023
Viewed by 1386
Abstract
Three-phase particulate composites offer greater design flexibility in the selection of phase materials and have more design variables than their two-phase counterparts, thus providing larger space for tailoring effective properties to meet intricate engineering requirements. Predicting effective elastic properties is essential for composite [...] Read more.
Three-phase particulate composites offer greater design flexibility in the selection of phase materials and have more design variables than their two-phase counterparts, thus providing larger space for tailoring effective properties to meet intricate engineering requirements. Predicting effective elastic properties is essential for composite design. However, experimental methods are both expensive and time intensive, whereas the scope of analytical micromechanics models is limited by their inherent assumptions. The newly developed microstructure-free finite element modeling (MF-FEM) approach has been demonstrated to be accurate and reliable for two-phase particulate composites. In this study, we investigate whether the MF-FEM approach can be applied to three-phase particulate composites and, if applicable, under which conditions. The study commences with a convergence analysis to establish the threshold ratio between the element size and the RVE (representative volume element) dimension. We then validate the MF-FEM approach using experimental data on three-phase composites from the existing literature. Subsequently, the MF-FEM method serves as a benchmark to assess the accuracy of both traditional and novel analytical micromechanics models, in predicting the effective elasticity of two distinct types of three-phase particulate composites, characterized by their small and large phase contrasts, respectively. We found that the threshold element-to-RVE ratio (1/150) for three-phase composites is considerably smaller than the ratio (1/50) for two-phase composites. The validation underscores that MF-FEM predictions align closely with experimental data. The analytical micromechanics models demonstrate varying degrees of accuracy depending on the phase volume fractions and the contrast in phase properties. The study indicates that the analytical micromechanics models may not be dependable for predicting effective properties of three-phase particulate composites, particularly those with a large contrast in phase properties. Even though more time-intensive, the MF-FEM proves to be a more reliable approach than the analytical models. Full article
(This article belongs to the Special Issue Finite Element Modeling of Microstructures in Composite Materials)
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Figure 1

Figure 1
<p>Composite RVE and coordinate system.</p>
Full article ">Figure 2
<p>RVE with 30%, 35%, and 35% of softer, stiffer, and intermediate phases (red, blue, purple) for different element-to-RVE size ratios (<b>a</b>) 1/50; (<b>b</b>) 1/100; (<b>c</b>) 1/150.</p>
Full article ">Figure 3
<p>Variation in RVE properties of composite SPC303535 with element-to-RVE size ratio: (<b>a</b>) Young’s modulus; (<b>b</b>) Poisson’s ratio.</p>
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<p>Variation in RVE properties of composite LPC303535 with element-to-RVE size ratio: (<b>a</b>) Young’s modulus; (<b>b</b>) Poisson’s ratio.</p>
Full article ">Figure 5
<p>Validation of MF-FEM against experimental data obtained by Cohen and Ishai [<a href="#B20-materials-16-06147" class="html-bibr">20</a>] (weight ratio <span class="html-italic">n</span> = 0.5:1).</p>
Full article ">Figure 6
<p>Validation of MF-FEM against experimental data by Yang [<a href="#B22-materials-16-06147" class="html-bibr">22</a>].</p>
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<p>Prediction of effective properties of three-phase composites using analytical micromechanics models.</p>
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<p>Bound gap for iterative isotropization of VR/HS model.</p>
Full article ">Figure 9
<p>Comparison of MF-FEM and analytical models against experimental data by Yang [<a href="#B21-materials-16-06147" class="html-bibr">21</a>]: (<b>a</b>) Young’s modulus prediction; (<b>b</b>) relative error in effective Young’s modulus calculation.</p>
Full article ">Figure 10
<p>Effective Young’s moduli of SPC composites characterized by MF-FEM and analytical micromechanics models with (<b>a</b>) 0%; (<b>b</b>) 20%; (<b>c</b>) 40%; (<b>d</b>) 50%; (<b>e</b>) 60%; and (<b>f</b>) 80% of the softer phase.</p>
Full article ">Figure 11
<p>Effective shear moduli of SPC composites characterized by MF-FEM and analytical micromechanics models with (<b>a</b>) 0%; (<b>b</b>) 20%; (<b>c</b>) 40%; (<b>d</b>) 50%; (<b>e</b>) 60%; and (<b>f</b>) 80% of the softer phase.</p>
Full article ">Figure 11 Cont.
<p>Effective shear moduli of SPC composites characterized by MF-FEM and analytical micromechanics models with (<b>a</b>) 0%; (<b>b</b>) 20%; (<b>c</b>) 40%; (<b>d</b>) 50%; (<b>e</b>) 60%; and (<b>f</b>) 80% of the softer phase.</p>
Full article ">Figure 12
<p>Effective bulk moduli of SPC composites characterized by MF-FEM and analytical micromechanics models with (<b>a</b>) 0%; (<b>b</b>) 20%; (<b>c</b>) 40%; (<b>d</b>) 50%; (<b>e</b>) 60%; and (<b>f</b>) 80% of the softer phase.</p>
Full article ">Figure 13
<p>Effective Poisson’s ratios of SPC composites characterized by MF-FEM and analytical micromechanics models with (<b>a</b>) 0%; (<b>b</b>) 20%; (<b>c</b>) 40%; (<b>d</b>) 50%; (<b>e</b>) 60%; and (<b>f</b>) 80% of the softer phase.</p>
Full article ">Figure 14
<p>Effective Young’s moduli of LPC composites characterized by MF-FEM and analytical micromechanics models with (<b>a</b>) 0%; (<b>b</b>) 20%; (<b>c</b>) 40%; (<b>d</b>) 50%; (<b>e</b>) 60%; and (<b>f</b>) 80% of the softer phase.</p>
Full article ">Figure 15
<p>Effective shear moduli of LPC composites characterized by MF-FEM and analytical micromechanics models with (<b>a</b>) 0%; (<b>b</b>) 20%; (<b>c</b>) 40%; (<b>d</b>) 50%; (<b>e</b>) 60%; and (<b>f</b>) 80% of the softer phase.</p>
Full article ">Figure 16
<p>Effective bulk moduli of LPC composites characterized by MF-FEM and analytical micromechanics models with (<b>a</b>) 0%; (<b>b</b>) 20%; (<b>c</b>) 40%; (<b>d</b>) 50%; (<b>e</b>) 60%; and (<b>f</b>) 80% of the softer phase.</p>
Full article ">Figure 16 Cont.
<p>Effective bulk moduli of LPC composites characterized by MF-FEM and analytical micromechanics models with (<b>a</b>) 0%; (<b>b</b>) 20%; (<b>c</b>) 40%; (<b>d</b>) 50%; (<b>e</b>) 60%; and (<b>f</b>) 80% of the softer phase.</p>
Full article ">Figure 17
<p>Effective Poisson’s ratios of LPC composites characterized by MF-FEM and analytical micromechanics models with (<b>a</b>) 0%; (<b>b</b>) 20%; (<b>c</b>) 40%; (<b>d</b>) 50%; (<b>e</b>) 60%; and (<b>f</b>) 80% of the softer phase.</p>
Full article ">Figure A1
<p>Relative errors in effective Young’s moduli of SPC composites with: (<b>a</b>) 0%; (<b>b</b>) 20%; (<b>c</b>) 40%; (<b>d</b>) 50%; (<b>e</b>) 60%; and (<b>f</b>) 80% of softer phase.</p>
Full article ">Figure A2
<p>Relative errors in effective shear moduli of SPC composites with: (<b>a</b>) 0%; (<b>b</b>) 20%; (<b>c</b>) 40%; (<b>d</b>) 50%; (<b>e</b>) 60%; and (<b>f</b>) 80% of softer phase.</p>
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<p>Relative errors in effective bulk moduli of SPC composites with: (<b>a</b>) 0%; (<b>b</b>) 20%; (<b>c</b>) 40%; (<b>d</b>) 50%; (<b>e</b>) 60%; and (<b>f</b>) 80% of softer phase.</p>
Full article ">Figure A4
<p>Relative errors in effective Poisson’s ratios of SPC composites with: (<b>a</b>) 0%; (<b>b</b>) 20%; (<b>c</b>) 40%; (<b>d</b>) 50%; (<b>e</b>) 60%; and (<b>f</b>) 80% of softer phase.</p>
Full article ">Figure A4 Cont.
<p>Relative errors in effective Poisson’s ratios of SPC composites with: (<b>a</b>) 0%; (<b>b</b>) 20%; (<b>c</b>) 40%; (<b>d</b>) 50%; (<b>e</b>) 60%; and (<b>f</b>) 80% of softer phase.</p>
Full article ">Figure A5
<p>Relative errors in effective Young’s moduli of LPC composites with: (<b>a</b>) 0%; (<b>b</b>) 20%; (<b>c</b>) 40%; (<b>d</b>) 50%; (<b>e</b>) 60%; and (<b>f</b>) 80% of softer phase.</p>
Full article ">Figure A6
<p>Relative errors in effective shear moduli of LPC composites with: (<b>a</b>) 0%; (<b>b</b>) 20%; (<b>c</b>) 40%; (<b>d</b>) 50%; (<b>e</b>) 60%; and (<b>f</b>) 80% of softer phase.</p>
Full article ">Figure A7
<p>Relative errors in effective bulk moduli of LPC composites with: (<b>a</b>) 0%; (<b>b</b>) 20%; (<b>c</b>) 40%; (<b>d</b>) 50%; (<b>e</b>) 60%; and (<b>f</b>) 80% of softer phase.</p>
Full article ">Figure A8
<p>Relative errors in effective Poisson’s ratios of LPC composites with: (<b>a</b>) 0%; (<b>b</b>) 20%; (<b>c</b>) 40%; (<b>d</b>) 50%; (<b>e</b>) 60%; and (<b>f</b>) 80% of softer phase.</p>
Full article ">
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